Can the mean squared error be negative?
In general a cost function can be negative. The more negative, the better of course, because you are measuring a cost the objective is to minimise it.

A standard Mean Squared Error function cannot be negative. The lowest possible value is $0$, when there is no output error from any example input.


  How can our cost function which is mean squared error have a value under 0?


It cannot. You don't link the precise graph or lecture where you saw this, but I would suspect Andrew Ng drew a representative graph for any cost function in order to point out that it would typically have an optimum, minimum value. He may have been talking at the same time about MSE as an example.

Many loss or cost functions are designed with an absolute minimum of $0$ possible for "no error" results. In supervised learning that is often a simple consequence of basing the cost on the difference between the model outputs and desired outputs. So in supervised learning problems of regression and classification, you will rarely see a negative cost function value. But there is no absolute rule against negative costs in principle.